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Negative and Positive Angles Cutting a Circle

The circle with its center at the origin is a platform for describing all the possible angle measures from 0 to 360 degrees, plus all the negatives of those angles, and plus all the multiples of the positive and negative angles from negative infinity to positive infinity. In other words, the unit circle consists of all the angles that exist. Because a right triangle can only measure angles of 90 degrees or less, the circle allows for a much-broader range.

Positive angles on a unit circle

The positive angles on the unit circle are measured with the initial side on the positive x-axis and the terminal side moving counterclockwise around the origin.

Several positive angles on the unit circle.
Several positive angles on the unit circle.

The figure shows some positive angles measured in both degrees and radians. Notice that putting together the terminal sides of the angles measuring 30 degrees and 210 degrees, or putting together the terminal sides of angles measuring 60 degrees and 240 degrees, and so on gives you straight lines. This fact is to be expected because the angles are 180 degrees apart, and a straight angle measures 180 degrees.

Negative angles on a unit circle

Just when you thought that 360 degrees or 2π radians was enough for anyone, you’re confronted with the reality that many of the basic angles have negative values and even multiples of themselves. If you measure angles clockwise instead of counterclockwise, then the angles have negative measures: a 30-degree angle is the same as a –330-degree angle, because they have the same terminal side. Likewise, an angle of 7π/3 is the same as an angle of –π/3.

But wait — you have even more ways to name an angle. By doing a complete rotation or two (or more) and adding or subtracting 360 degrees or a multiple of it before settling on the angle’s terminal side, you can get an infinite number of angle measures, both positive and negative, for the same basic angle. For example, an angle of 60 degrees has the same terminal side as that of a 420-degree angle and a –300-degree angle. The preceding figure shows many names for the same 60-degree angle in both degrees and radians.

Although this name-calling of angles may seem pointless at first, there’s more to it than arbitrarily using negatives or multiples of angles just to be difficult. The angles that are related to one another have trig functions that are also related, if not the same.

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